UC-NRLF 


EXCHANGE 


EXCHANGE 

FFR   28  1P17 


Activity   and    Concentration,    Transport 
Numbers  and  Boundary  Potential 


A  DISSERTATION 


SUBMITTED  IN  PARTIAL  FULFILMENT  OF  THE  REQUIREMENTS 

FOR    THE    DEGREE    OF    DOCTOR   OF    PHILOSOPHY    IN 

THE  UNIVERSITY  OF  MICHIGAN,  JUNE,  1915. 


By 


ALFRED  LYNN  FERGUSON 


EASTON,  PA.: 

ESCHENBACH  PRINTING  COMPANY 
1916 


Activity   and    Concentration,    Transport 
Numbers  and  Boundary  Potential 


A  DISSERTATION 


SUBMITTED  IN   PARTIAL  FULFILMENT  OF  THE  REQUIREMENTS 

FOR    THE    DEGREE    OF    DOCTOR    OF    PHILOSOPHY    IN 

THE  UNIVERSITY  OF  MICHIGAN,  JUNE,  1915. 


By 
ALFRED  LYNN  FERGUSON 


E ASTON,  PA.: 

ESCHENBACH  PRINTING  COMPANY 
1916 


The  writer  wishes  to  express  his  earnest  appreciation  for 
the  kind  advice  and  helpful  assistance  rendered  by  Professor 
R.  C.  Tolman,  under  whose  direction  some  preliminary  work 
was  carried  out,  without  which  the  present  investigation  would 
not  have  been  possible. 

It  is  with  sincere  gratitude  that  acknowledgment  is  here 
made  to  Professor  S.  L.  Bigelow  for  the  interest  he  has  taken 
in  this  work  and  for  his  most  valuable  criticisms. 


TABLE  OF  CONTENTS. 

Introduction 5 

Theoretical .' 6 

Apparatus  and  Materials 12 

Method  of  Procedure  and  Experimental  Results. 18 

Discussion  of  Results 21 

Conclusion 31 


ACTIVITY     AND     CONCENTRATION,     TRANSPORT 
NUMBERS,  AND  BOUNDARY  POTENTIAL1 


Introduction 

In  this  investigation  of  the  relation  between  activity  and 
concentration,  a  new  form  of  cell  was  used  which  permits 
of  two  new  methods  for  measuring  transport  numbers  and 
a  direct  method  for  measuring  boundary  electromotive  force. 

There  are  various  types  of  concentration  cells,  but  the 
type  which  has  excited  the  most  interest  and  stimulated  the 
greatest  amount  of  experimental  and  theoretical  investigation 
is  that  one  in  which  the  electrodes  are  of  the  same  material, 
but  dip  into  solutions  of  different  concentrations  of  the  same 
electrolyte.  An  extensive  investigation  of  such  cells  was 
carried  out  by  Hans  Jahn.  His  primary  object  was  to  test 
the  existing  formulas  and  theories  of  solutions  when  applied 
to  dilute  and  moderately  concentrated  solutions  of  strong 
electrolytes.  From  his  work  Jahn  felt  justified  in  concluding: 
i/(i)  that  conductivity  measurements  do  not  give  satisfactory 
values  for  the  degree  of  dissociation;  and  (2)  that  the  rnobility 
of  the  ions  is  a  function  of  the.  concentration  increasing  with 
increasing  concentration.  These  are  remarkable  conclusions 
and  constitute  a  blow  aimed  directly  at  the  formula  a.  = 
M»/Moo°f  Arrhenius  for  calculating  the  degree  of  dissociation. 
Naturally  they  provoked  much  discussion  and  criticism.  The 
principal  participants  in  this  discussion  were  H.  Jahn,2  Arrhe- 
nius,3 Nernst,4  Lehfeldt,5  Goebel,6  Sand,7  Planck,8  and  Kruger.9 


1  Contribution    from    the    Chemical    Laboratory    of    the    University    of 
Michigan. 

2  Zeit.  phys.  Chem.,  35,  i  (1900);  36,  453;  37,  490;  38,  125  (1901);  4*1  25? 
(1907). 

3  Ibid.,  36,  28;  37,  315  (1901). 

4  Ibid.,  36,  596;  38,  487  (1901). 
6  Ibid.,  35,  257  (1900). 

6  Ibid.,  42,  54  (1902). 

7  Ibid.,  36,  499  (1901). 

8  Ibid.,  41,  212  (1902). 

9  Ibid.,  36,  86  (1901). 


As  a  result  of  this  discussion,  Jahn  was  forced  to  admit 
several  mistakes  in  his  reasoning,  but  he  maintained  to  the 
end  that  the  above  stated  conclusions  were  justifiable.  For 
a  while,  however,  his  work  was  rather  discredited,  but  within 
the  past  fifteen,  and  particularly  within  the  last  five  years, 
results  have  been  obtained  supporting  his  conclusions:  Such 
for  instance,  as  the  investigation  of  F.  Flugel1  on  the  freezing 
point  lowering  and  conductivity  of  very  dilute  solutions. 
The  study  of  concentration  cells  with  thalium  electrodes  by 
Lewis  and  Von  Ende.2  The  investigation,  by  Lewis  and  Ed- 
gar,3 of  the  equilibrium  between  nitric  acid,  nitrous  acid,  and 
nitric  oxide.  The  influence  of  one  salt  on  the  solubility  of 
another  as  worked  out  by  Bray.4  The  work  of  Washburn  and 
Maclnnes5  on  the  freezing  point  lowering  and  conductivity 
of  cesium  nitrate.  And  an  earlier  investigation  by  the  author 
in  conjunction  with  R.  C.  Tolman6  on  the  free  energy  of  dilu- 
tion of  hydrochloric  acid. 

In  this  earlier  investigation  it  was  concluded  that  the  ratio 
of  concentrations  of  ions  was  in  every  instance  greater  than 
the  ratio  of  their  activities,  while  the  concentration  ratios  for 
the  undissociated  acid  were  in  every  case  less  than  the  ac- 
tivity ratios.  The  variation  was  found  to  be  much  greater 
for  the  undissociated  acid  than  for  the  ions.  In  all  these  cal- 
culations one  concentration  was.  tenth  normal  and  the  other 
was  something  less.  A  form  of  cell  was  used  that  eliminated 

transport  numbers. 

Theoretical 

The  application  of  thermodynamic  theory  to  physico- 
chemical  measurements  is  rapidly  increasing  in  importance 
and  nowhere  is  it  more  important  than  in  connection  with 
such  cells  as  form  the  subject  of  this  investigation. 


1  Zeit.  phys.  Chem.,  79,  577  (1912). 

2  Jour.  Am.  Chem.  Soc.,  32,  737  (1910). 

3  Ibid.,  33,  292  (1911). 

4  Ibid.,  33,  1673  (1911). 

5  Ibid.,  33,  1686  (1911). 

6  Ibid.,  34,  232  (1912). 


The  work  which  a  reversible  cell  is  capable  of  furnishing 
must  equal  its  electromotive  force  times  the  quantity  of  elec- 
tricity passing  through  the  cell.  That  is,  w  =  EF  per  equiva- 
lent, where  F  equals  one  Faraday  and  E  equals  electromotive 
force.  • 

Suppose  we  consider  a  hydrochloric  acid  concentration 
cell  with  calomel  electrodes  and  the  acid  on  the  two  sides  having 
the  concentrations  Ci  and  C2.  When  one  equivalent  of  elec- 
tricity passes  through  the  cell,  the  following  changes  take  place : 
On  the  concentrated  side,  one  equivalent  of  Cl  ion  will  be 
removed  to  combine  with  the  equivalent  of  Ag  ion  formed; 
Na  equivalents  of  Cl  ion  will  migrate  in  from  the  dilute  side, 
where  Nfl  represents  the  transport  number  for  the  Cl  ion. 
On  the  dilute  side  one  equivalent  of  Cl  ion  will  form;  N0 
equivalent  of  Cl  ion  will  migrate  out  to  the  concentrated  side, 
and  i  -  -  N0  equivalents  of  H  ion  will  migrate  in  from  the  con- 
centrated side.  The  net  result  is  then  the  disappearance  of 
Nfl  equivalents  of  HC1  from  the  concentrated  side  and  the  ap- 
pearance of  N0  equivalents  on  the  dilute  side,  or,  in  other  words, 
the  transfer  of  Na  equivalents  of  HC1  from  one  concentration 
to  another.  The  electromotive  force  of  such  a  cell  multiplied 
by  one  Faraday  gives  the  free  energy  of  dilution  of  N0  mols 

RF 
of  HC1,  so  the  free  energy  per  mol  is  M   .      This  is  the  type  of 

IN  a 

cell  used  by  Jahn  in  his  measurements  of  free  energy.  The 
objections  to  this  type  of  cell  have  been  pointed  out  in  an  earlier 
paper.1 

Let  us  now  consider  the  conception  developed  by  Lewis2, 
which  he  calls  activity.  This  quantity  is  defined  by  the  fol- 
lowing conditions: 

1.  "The  activity  of  a  molecular  species  is  the  same  in 
two  phases  when  these  phases  are  in  equilibrium  as  regards 
the  distribution  of  that  species." 

2.  "The  activity  of  a  gas  approaches  the  gas  concentra- 
tion as  a  limiting  value  if  the  gas  is  indefinitely  rarefied.'* 

1  Loc.  cit. 

2  Proc.  Am.  Acad.,  43,  257  (1907). 


8 

In  an  earlier  paper1  he  developed  another  quantity  which  he 
called  "escaping  tendency." 

In  yet  another2  he  uses  the  same  conception  but  calls  it, 
for  brevity,  "fugacity."  The  quantity  fugacity  has  the  di- 
mensions of  pressure  while  activity  has  the  dimensions  of  con- 
centration, and  the  two  are  connected  by  the  formula  a  = 

,   where  a  represents    activity,    ^    fugacity,    R    the    gas 


constant  and  T  the  absolute  temperature. 

Let  us  apply  this  activity  conception  to  a  hydrochloric 
acid  concentration  cell.  If  the  activities  of  undissociated 
HC1  in  the  two  solutions  be  represented  by  a'Hci  and  a"HCi 
and  the  activities  of  the  ions  by  a'n,  a"^,  a'^,  a"~^  then  by 
substituting  in  the  equation  for  free  energy 

F  =  KF  =  RT  In  - 
Oi 

we  have  for  the  free  energy  change,  accompanying  the  trans- 
fer of  one  mol  of,,HCl  from  one  solution  to  the  other 

F  =  EF  =  RT  In  =  RT  In  a'*  X  ° 


. 
a  HC1  a  H  X  a  Ci 

We  may  ass-ume  also  that  a'J   =  a'ci  and  that  a"n  =  a"ci> 
then 

EF  =  2RT  In  ^  =  2RT  In  ^   =  RT  In  ^.      (i) 

a  H  a  Cl  a  HC1 

This  equation  applies  to  concentration  cells  without  boundary 
potential. 

For  concentration  cells  having  boundary  potential,  and 
reversible  with  respect  to  the  anion,  we  have  the  well  known 
formula, 

NCRT       a"£ 

Hi  =  2  -—  In  -ft.  .      (2) 

a  H 

If  the  cell  is  reversible  with  respect  to  the  cation  the  formula 
becomes 


1  Proc.  Am.  Acad.,  36,  145  (1900). 

2  Ibid.,  37,  49  (1901). 


_,  NaRT       a"c, 

H2  =  2  — —  /n     ,_  .  (3) 

F  a'ci 

T? 
A  combination  of  (i)  and  (2)  gives  Nc  =  p1- 

TJ 

A  combination  of  (i)  and  (3)  gives  Na  =  -g2- 

Since  Nc  -f-  Na  =  i,  we  have  a  very  good  check  on  our 
work,  as  both  of  these  can  be  determined.  Also,  since  N^  + 

N«  =    El  ^  ^  =  i,  it  is  evident  that  Hi  +  E2  =  E. 

A  consideration  of  these  equations  shows  that  all  that  is 
necessary,  then,  to  measure  the  transference  number  of  the 
hydrogen  ion  is  to  measure  the  electromotive  force  of  a  con- 
centration cell  with  transference  and  without  using  hydrogen 
electrodes.  The  transference  number  of  the  chloride  ion 
in  the  same  solution  could  be  determined  in  a  similar  way 
by  using  calomel  electrodes. 

It  is  a  well  known  fact  that  the  surface  of  contact  between 
two  solutions  is  a  source  of  potential  difference.  This  po- 
tential difference  between  solutions  has  been  one  of  the  greatest 
stumbling  blocks  in  the  way  of  potential  measurements  of 
concentration  cells.  Those  who  have  made  such  measurements 
have  attempted  to  avoid  it  in  one  way  or  another  or  dismissed 
it  entirely  with  the  statement  that  it  was  too  small  to  consider. 

The  first  investigator  to  give  a  reasonable  explanation 
for  this  liquid  potential  was  Nernst.1  He  based  his  explana- 
tion on  the  unequal  velocities  of  migration  of  the  ions.  Sup- 
pose, for  example,  a  concentrated  and  a  dilute  solution  of  hydro- 
chloric acid  are  in  contact.  The  hydrogen  ions  and  the  chloride 
ions  of  the  concentrated  solutions  diffuse  into  the  dilute 
side.  But  the  hydrogen  ions  travel  with  a  much  greater  ve- 
locity than  the  chloride  ions  and,  since  they  carry  positive 
charges  and  the  chloride  ions  negative  charges,  the  dilute 
solution  becomes  positive  with  reference  to  the  concentrated. 
This  separation  cannot  take  place  to  any  appreciable  extent, 
however,  because  of  the  attraction  between  these  electrostatic 


1  Zeit.  phys.  Chem.,  4,  129  (1889). 


IO 


charges.  The  result  of  this  attraction  is  to  increase  the  velocity 
of  the  chloride  ions  and  lessen  the  velocity  of  the  hydrogen 
ions  till  eventually  both  travel  with  the  same  velocity,  with 
the  hydrogen  in  the  lead;  thus  an  electrical  double  layer  is 
formed  with  its  accompanying  difference  in  potential.  Nernst 
not  only  gave  this  explanation  for  the  phenomena  but  he  also 
derived  a  formula  for  calculating  the  values  of  such  potential 
differences.  With  activity  substituted  for  concentration  the 
formula  derived  by  him  takes  the  form 

E=    ^-->N°)RT^'. 

r  a% 

A  consideration  of  the  HC1  concentration  cell,  reversible 
with  respect  to  the  chloride  ion,  shows  that  the  boundary 
potential  acts  in  the  same  direction  as  the  electrode  potentials, 
while  in  the  HC1  concentration  cell,  reversible  with  respect 
to  the  hydrogen  ion,  the  boundary  potential  is  opposed  to  the 
electrode  potentials.  Thus  it  is  evident  that  this  boundary 
potential  is  equal  to  one-half  the  difference  between  the  elec- 
tromotive '  forces  of  such  concentration  cells,  and,  therefore, 
the  measurement  of  contact  potential  between  solutions 
differing  only  in  concentration  of  the  same  electrolyte  resolves 
itself  simply  into  the  simultaneous  measurement  of  the  elec- 
tromotive force  between  electrodes  which  furnish  the  positive 
ions  in  the  solution  and  the  electromotive  force  between  elec- 
trodes which  furnish  the  negative  ions.  Or,  in  the  case  of 
hydrochloric  acid,  the  measurement  of  the  electromotive 
force  of  the  hydrogen  concentration  cell  and  at  the  same  time 
the  measurement  of  the  electromotive  force  of  the  calomel 
concentration  cell. 

The  formula  for  the  electromotive  force  of  a  concentration 
cell  without  boundary  potential  shows  that  the  only  variables 
are  the  activity  ratio  and  the  electromotive  force.  If  one  of 
these  is  measured  the  other  may  be  calculated.  The  concen- 
tration ratio  can  be  calculated  from  conductivity  measure- 
ments. The  expression  for  the  ratio  of  ion  concentrations  is 

—r—  ,  where  a"  and  a'  are  the  degrees  of  dissociation 


II 


at  normalities  n"  and  n' .     The  ratio  of  undissociated  acid  is 

C"HCI    _  n"(l  — a"} 

As  has  been  stated,  the  conclusion  was  drawn  in  an  earlier 
paper  that  the  ratios  of  ion  concentrations  calculated  from 
conductivity  measurements  were  in  every  instance  greater  than 
the  measured  ratios  of  activities,  -while  the  ratios  for  the 
undissociated  acid  were  in  every  case  less  than  the  activity 
ratios.  Many  have  realized  that  this  great  variation  between 
observed  and  calculated  values  should  be  more  thoroughly 
investigated.  These  discrepancies  between  the  activity  ratios 
and  concentration  ratios  for  both  the  ions  and  un-ionized  acid 
can  be  accounted  for  in  either  of  two  ways.  It  may  be  that 
conductivity  measurements  give  the  true  value  for  the  degree 
of  dissociation,  in  which  case  activity  and  concentration  are 
not  proportional.  Or,  it  may  be  that  activity  and  concen- 
tration are  proportional,  in  which  case  conductivity  measure- 
ments can  no  longer  be  considered  a  true  hieasure  for  ion  con- 
centration. The  fact  that  activity  and  concentration  have 
been  shown  to  be  proportional  for  all  ordinary  solutions  of 
nonelectrolytes  and  even  for  weak  electrolytes  makes  the  second 
of  the  above  alternatives  the  more  probable.  In  view  of  this, 
a  careful  investigation  of  the  basis  for  Arrhenius'  formula, 

u 

a  =         'is  likely  to  reveal  something  of  interest. 

Moo 

The  conductivity  of  a  liquid  is  the  current  density  under 
unit  potential  gradient  and  may  be  represented  by 

k  =  FaC(U«  +  Ue), 

where  F  represents  one  Faraday,  a  the  degree  of  dissociation, 
C  the  total  concentration  in  gram  equivalents  per  cubic  centi- 
meter, U0  the  mobility  of  the  anion  and  Uc  the  mobility  of 
the  cation. 

If  this  specific  conductivity  is  multiplied  by  the  volume 
V  containing  one  equivalent,  it  gives  the  equivalent  conduc- 
tivity which  may  then  be  represented  by 

M  =  FaCV(Ufl  +  Ue)  =  Fa(Ua  +  Uc). 


12 


As  the  dilution  increases,  it  is  an  observed  experimental 
fact  that  a  increases  up  to  a  certain  limiting  value  characteristic 
for  each  electrolyte.  According  to  the  dissociation  theory, 
the  explanation  for  this  is  that  the  dissociation  becomes  greater 
the  greater  the  dilution  up  to  a  certain  limit  for  each  elec- 
trolyte where  a  =  i  .  The  formula  for  the  equivalent  conduc- 
tivity at  this  point  of  complete  dissociation  would  then  be- 
come M  =  F(Ufl  +  Uc).  By  a  combination  of  this  formula 
with  the  one  for  the  equivalent  conductivity  at  any  dilution, 
V,  there  is  obtained  an  expression 

C  -f  Ua)        a(Uc  +  Ua) 


Moo    "      F(UC  +  U«)  (VC  +  U0)  ' 

In  the  numerator  of  this  fraction,  Ua  and  Uc  represent  the 
mobilities  of  cation  and  anion  at  the  dilution  V,  while  in  the 
denominator  the  same  terms  represent  the  mobilities  at  infinite 
dilution.  It  is  evident  then,  in  order  that  the  formula  of 
Arrhenius  shall  be  true,  that  the  mobilities  of  the  ions  must 
be  independent  of  the  concentration.  It  is  this  assumption 
that  introduces  the  greatest  doubt  into  this  method  for  cal- 
culating ion  concentration. 

It  was  with  the  object  of  determining  whether  the  ions 
change  their  mobility  with  concentration,  that  the  author 
devised  the  method  used  in  this  work.  This  method  consists 
in  a  combination  of  electromotive  force  measurements  of  con- 
centration cells  with  contact  potential,  and  without  contact 
potential. 

Apparatus  and  Materials 

The  Cells.  —  A  variety  of  cells  was  tried  out  before  an 
entirely  suitable  one  was  found.  The  nature  of  the  problem 
made  it  desirable  to  be  able  to  take  series  of  measurements 
on  four  different  combinations  from  the  same  set-up.  These 
measurements  were  as  follows:  (i)  The  electromotive  force 
between  a  hydrogen  and  calomel  electrode  in  HC1  of  concen- 
tration Ci.  (2)  A  similar  measurement  but  with  HC1  of  con- 
centration C2.  (3)  The  electromotive  force  between  the  calomel 
electrode  in  concentration  Ci  and  the  one  in  C2.  (4)  A  similar 


13 

measurement  for  the  hydrogen  electrodes.  From  (i)  could 
be  calculated  the  free  energy  of  formation  of  HC1  from  hydrogen 
and  calomel  in  concentration  Ci.  From  (2)  a  similar  calcula- 
tion for  the  case  where  the  concentration  is  C2.  From  the 
difference  between  (i)  and  (2)  could  be  obtained  the  free  energy 
of  dilution  of  HC1  from  one  concentration  to  the  other.  The 
difference  between  (i)  and  (2)  would  give  also  the  E  of  Equa- 
tion i.  The  measurements  in  (3)  would  give  the  electro- 
motive force  of  the  concentration  cell  having  boundary  po- 
tential with  calomel  electrodes,  or  the  value  of  Ei  in  Equation 
2.  Similarly  (4)  would  give  E2  in  Equation  3. 

The  above  data  would  make  it  possible  to  calculate  the 

ratios     ,5l,  —£  and  ~-f  HC1 ;    also    the    transference    numbers 
a  ci     a  H  a  HCI 

for  both — the  hydrogen  and  chloride  ions.  Several  other 
things  had  to  be  considered  in  the  construction  of  these 
cells  in  order  to  be  sure  that  the  above  electromotive  force 
values  represented  accurately  what  they  were  supposed 
to.  First  of  all,  they  should  remain  constant  a  reasonable 
length  of  time.  This  necessitated  extreme  caution  to  pre- 
vent diffusion  between  the  solutions  of  different  concentrations. 
As  a  result  of  an  extensive  study  of  the  calomel  electrodes, 
it  was  decided  they  must  be  in  separate  chambers  from  the 
hydrogen  electrodes.  Slight  unavoidable  movements  of  the 
connecting  tube  when  this  passed  through  the  calomel  to  the 
mercury,  as  is  generally  the  case,  agitated  the  calomel  and 
mercury  in  D  (Fig.  i)  and  caused  a  change  in  thevalueof  theelec- 
trode.  To  avoid  this  source  of  error  it  was  thought  advisable 
to  make  contact  directly  with  the  mercury.  With  these  various 
requirements  in  mind  a  set-up  of  the  type  shown  in  Fig.  i 
was  finally  used.  Compartment  A  contains  a  calomel  elec- 
trode in  acid  of  concentration  C2 ;  B  contains  a  hydrogen  elec- 
trode in  acid  of  the  same  concentration;  C  contains  a  calomel 
electrode  in  acid  of  concentration  d,  and  D  contains  a  hydrogen 
electrode  in  acid  of  this  same  concentration.  The  various 
compartments  are  connected  through  the  three-way  stopcocks 
a,  b,  c  and  d.  The  rubber  connections  between  B  and  D; 


and  A  and  C  contain  filter  paper  to  reduce  the  diffusion; 
also  the  holes  of  the  three-way  stopcocks  in  b  and  d  contain 
filter  paper.  The  hydrogen  inlet  tubes  are  at  e  and  /,  and  the 
outlet  tubes  at  g  and  h.  In  each  hydrogen  compartment 
are  two  electrodes  i  and  i' ,  and  /  and  /'. 


The  Hydrogen  Electrodes. — An  extended  study  of  the  hy- 
drogen electrodes  was  made  to  settle  the  following  points: 
the  size  and  shape  most  suitable  for  our  cells,  ease  of  construc- 
tion, effect  of  varying  amounts  of  platinum  black,  the  constancy 
when  changed  from  a  solution  of  one  concentration  to  a  solu- 
tion of  another  concentration,  reproducibility,  and  effect  of 
exposure  to  air  for  several  days.  This  study  showed  that  the 


15 

size  and  shape  of  the  electrodes  has  no  influence  on  their  value. 
Those  finally  used  in  this  work  were  made  of  platinum  foil 
about  one  and  one-half  centimeters  by  two  centimeters.  As 
long  as  the  coating  had  a  black  velvety  appearance  the  elec- 
trodes remained  constant;  but  after  continued  use  they 
turned  gray,  became  less  sensitive,  and  fluctuated  in  value. 
The  same  electrodes  were  used  several  times  before  replatin- 
izing.  It  was  found  inadvisable  to  have  too  much  platinum 
black.  As  to  constancy,  they  proved  to  be  perfectly  satis- 
factory. Electrodes  were  used  in  solutions  of  different  con- 
centrations, where  they  remained  for  several  days,  in  some 
instances,  and  when  they  were  returned  to  the  stock  solution 
they  were  found  to  have  suffered  no  change.  The  degree  of 
reproducibility  is  shown  by  the  fact  that  when  twelve  of  these 
electrodes  were  made  and  placed  in  a  stock  solution  of  very 
dilute  acid  and  eleven  were  compared  with  the  remaining 
one,  eleven  were  exactly  at  the  same  potential  and  the  twelfth 
showed  a  variation  of  only  o.oi  mv.  Over  a  period  of  eight 
days  the  greatest  variation  was  0.02  mv.  Some  of  these 
electrodes  were  exposed  to  the  air  for  several  days  without  any 
permanent  effect.  An  accident  which  happened  to  one  of 
them  shows  they  will  stand  much  abuse.  This  electrode  be- 
came unevenly  covered  with  paraffin,  and  was  left  on  a  shelf 
for  several  months.  To  remove  the  paraffin  the  electrode 
was  soaked  in  benzole,  and  to  remove  any  other  organic  matter 
it  was  boiled  in  nitric  acid  and  then  several  times  in  water. 
Again  it  was  put  through  the  platinizing  and  washing  process 
and  returned  to  the  stock  solution.  A  comparison  showed  it 
to  be  at  exactly  the  same  potential  with  the  standard.  This 
study  shows  that  these  hydrogen  electrodes  would  make  a 
better  standard  than  the  calomel  electrodes.  Nernst1 
was  the  first  one  to  recognize  the  advantages  of  the  hydrogen 
electrode  and  advocated  its  use  as  a  standard  in  preference 
to  the  calomel  electrode. 

The  Calomel  Electrodes. — An  investigation  of  the  calomel 
electrodes,  even  more  extensive  than  with  the  hydrogen  elec- 

1  Zeit.  Elektrochemie,  7,  253  (1900). 


i6 

trodes,  was  necessary  because  of  the  gradual  and  often  rapid 
changes  which  took  place  in  them. 

They  were  all  made  from  the  same  stock  of  Baker  and 
Adamson  analyzed  calomel  (Serial  no.  3772)  and  twice  dis- 
tilled mercury.  It  was  found  better  not  to  use  mercury  in 
making  the  paste.  The  calomel  was  mixed  in  a  small  beaker 
with  a  little  of  the  solution  to  be  used,  until  it  was  creamy, 
and  then  was  washed  three  times  by  decantation  with  the  same 
solution.  A  layer  of  calomel  of  considerable  depth  was  found 
advisable.  Connection  with  the  electrode  was  made  through 
the  side  arm  (k),  Fig.  i. 

The  first  cells  used  had  the  calomel  and  hydrogen  in  the 
same  compartment.  Two,  and  often  four,  of  these  were  made 
up  at  the  same  time.  In  a  few  minutes  they  would  come  to- 
gether within  a  few  hundred ths  of  a  millivolt.  Soon  after 
the  hydrogen  started  to  flow  through  the  cell,  the  calomel 
electrodes  began  slowly  to  change.  Some  of  the  solutions, 
after  there  had  been  considerable  change  in  the  values  of  the 
cells,  were  titrated  against  barium  hydroxide,  and  in  every 
case  increase  in  strength  of  acid  was  observed.  The  reason 
for  this  change  in  concentration  has  not  yet  been  determined. 
This  increase  was  roughly  (not  exactly)  proportional  to  the 
change  which  had  taken  place  in  the  electromotive  force  of 
the  cell.  Solutions  taken  from  cells  that  remained  constant 
gave  the  same  analysis  as  the  original  solution.  It  was  ob- 
served, moreover,  that  in  the  cells  which  had  changed,  the 
surface  of  the  calomel  appeared  darker.  In  other  cases,  when 
freshly  platinized  electrodes  were  used,  a  few  small  specks  of 
platinum  black  became  loosened  from  the  electrode  by  the 
action  of  the  bubbles  of  hydrogen.  In  these  cases  the  rate 
of  change  of  electromotive  force  was  greater.  To  prevent 
these  particles  of  platinum  black  from  reaching  the  calomel, 
the  electrodes  were  covered  with  a  filter  paper  bag,  but  this 
did  not  prove  satisfactory.  These  facts  offered  an  explana- 
tion for  the  changes  that  had  been  observed.  There  appeared 
to  be  close  connection  between  the  change  in  the  electro- 
motive force  and  the  darkening  of  the  calomel.  Often  the 


17 

surface  became  nearly  black;  in  such  cases,  a  large  increase 
in  concentration  of  acid  was  found.  To  determine  whether 
simply  the  passage  of  hydrogen  through  the  cell  would  produce 
a  change  in  concentration,  a  sample  of  calomel  mercury  paste 
was  placed  in  each  of  two  tubes  and  covered  with  N/6o  acid. 
Through  one  of  these,  hydrogen  was  passed  for  two  days  while 
the  other  was  left  undisturbed.  No  change  in  color  of  the 
calomel  was  observed,  and  no  appreciable  change  in  concen- 
tration. These  results  are  not  sufficient  to  establish  any  con- 
clusion, but  they  are  enough  to  arouse  the  suspicion  that  the 
changes  observed  in  the  calomel  electrodes  are  due  to  a  change 
in  concentration  of  the  acid.  This  in  turn  is  in  some  way  con- 
nected with  the  combined  action  of  hydrogen,  platinum  black 
and  calomel.  It  might  be  stated,  as  a  possibility,  that  the 
hydrogen,  aided  by  the  catalytic  action  of  platinum  black, 
decomposes  the  calomel  with  the  formation  of  more  HC1 
and  free  mercury.  Much  more  work  would  be  required  be- 
fore any  definite  statement  could  be  made,  but  no  difficulties 
were  encountered  when  the  form  of  cell  in  Fig.  i  was  used. 

The  Thermostat. — In  all  earlier  work  the  ordinary  form 
thermostat  heated  by  a  gas  flame,  the  size  of  which  was  regu- 
lated by  the  rising  and  lowering  of  a  column  of  mercury, 
was  used.  This  did  not  prove  satisfactory.  The  variation 
in  gas  pressure,  and  the  corroding  of  the  mercury  surface  made 
close  attention  necessary  and  decreased  the  accuracy  of  regula- 
tion. To  avoid  these  difficulties  an  electrically  heated  and 
regulated  thermostat  was  constructed.  In  this  new  thermostat 
the  regulation  was  so  close  that  no  change  of  temperature 
could  be  detected  by  a  Beckman  thermometer  over  a  period 
of  several  days. 

The  Electrical  Apparatus. — The  electromotive  forces  were 
measured  with  an  Otto  Wolff  15,000  ohm  potentiometer  and 
suitable  galvanometer.  A  certified  Weston  cell  was  used  as 
a  standard  and  was  kept  in  the  same  thermostat  in  which  the 
measurements  were  carried  out. 

The  Hydrochloric  Acid. — An  approximately  fifth-normal 
stock  solution  of  hydrochloric  acid  (Baker  &  Adamson's 


i8 

analyzed  hydrochloric  acid  C.  P.)  was  prepared  by  dilution 
with  conductivity  water  and  its  concentration  determined  by 
precipitation  of  AgCl.  Five  analyses  showed  per  1000  grams 
solution,  7.259,  7.266,  7.262,  7.263  grams  of  chlorine.  The 
solutions  used  for  measurement  were  A/"/5>  N/io,  N/6o, 
N/go,  N/I20,  N/iSo,  and  ^300,  made  from  a  weighed  quan- 
tity of  the  stock  solution  by  dilution  at  18°,  using  flasks  care- 
fully calibrated  with  standardized  weights. 

The  Hydrogen  Generator. — The  hydrogen  used  was  gen- 
erated electrolytically  from  a  strong  sodium  hydroxide  solu- 
tion in  a  generator  of  the  type  described  by  Bodenstein.1 
The  electrodes  were  of  nickel  wire. 

In  order  to  remove  possible  traces  of  oxygen,  the  gas 
was  passed,  before  use,  through  a  tube  containing  a  platinum 
wire  heated  to  incandescence  by  an  electric  current.2  It  was 
then  passed  through  a  wash  bottle  filled  with  distilled  water, 
then  bubbled  from  a  fine  capillary  through  acid  of  the  same 
concentration  and  temperature  as  that  in  the  cell,  and  finally 
allowed  to  bubble  from  a  fine  capillary  through  the  solution 
surrounding  the  platinized  electrodes  and  escape  through  a 
long  tube  into  the  air. 

Method  of  Procedure  and  Experimental  Results 

In  all  of  the  measurements  N/$o  HC1  was  taken  as  the 
standard  for  comparison  and  they  were  all  made  at  18°  C.3 
The  cells  constructed  as  described  were,  in  most  cases,  set 
up  at  night,  and  by  the  next  night  the  readings  had  become 
constant.  Readings  were  taken  at  varying  intervals  over  a 
period  of  several  days. 

The  following  uniform  method  was  adopted:  The  stop- 
cock d,  Fig.  i,  in  the  tube  connecting  the  calomel  electrode 
in  concentration  C*  with  the  one  in  C3o  was  opened,  also  the 
similar  stopcock  b  between  the  hydrogen  electrodes.  After 


1  Zeit.  Elektrochemie,  u,  373  (1905). 

2  This  precaution  was  really  not  necessary  as  no  change  in  electromotive 
force  could  be  detected  when  the  wire  was  not  heated. 

3  The  bath  temperature  remained  so  constant  that  it  was  unnecessary 
to  record  it  in  the  tables. 


19 

about  five  minutes  the  potentials  were  measirred  and  the  stop- 
cocks closed.  By  producing  a  slight  suction  on  the  upward 
directed  tube  of  the  three-way  stopcock  a  connecting  the  hy- 
drogen chamber  and  calomel  chamber  of  the  N/^o  solution, 
the  liquid  in  the  downward  directed  branches  was  drawn  up 
till  contact  was  made  within  the  stopcock  a;  the  stopcock  / 
in  the  upward  tube  was  then  closed.  After  the  reading  was 
taken  I  was  opened  sufficiently  to  permit  a  separation  of  the 
liquid  in  a.  The  cell  was  left  in  this  condition  till  the  next 
reading.  A  similar  procedure  was  then  followed  for  the  cell 
containing  Cx  acid. 

The  experimental  data  obtained  in  this  work  are  included 
in  eighteen  tables  of  which  Table  I  in  this  article  is  a  sample. 
In  column  Eso  is  given  the  electromotive  force  between  the 
hydrogen  electrode  and  calomel  electrode  in  N/^o  HC1. 
Column  Eeo,  Ego,  — ,  — ,  etc.,  in  the  various  tables  contains 
the  electromotive  force  measurements  between  the  hydrogen 
and  calomel  electrodes  in  concentrations  N/6o,  N/qo,  — ,  — , 
etc.  Under  E-2  are  given  the  potentials  between  a  hydrogen 
electrode  in  N/^o  HC1  and  the  hydrogen  electrode  in  the  other 
concentration  being  investigated ;  and  under  Ei  are  given  corre- 
sponding values  for  the  calomel  electrodes. 

As  was  stated  in  the  discussion  above,  E*  —  Eso  should 
equal  Ei  +  £2-  The  remarkable  agreement  found  is  shown  by 
a  comparison  of  the  last  two  columns. 

The  slight  variations  observed  between  successive  read- 
ings can  be  accounted  for  largely  by  changes  in  barometric 
pressure.1  The  effect  of  these  changes  is  practically  the  same 
in  acid  of  all  concentrations,  and  as  we  are  concerned  with 
differences  only,  it  is  not  necessary  to  make  corrections  for  them. 

A  large  number  of  preliminary  experiments  were  per- 
formed before  all  of  the  disturbing  elements  were  discovered 
and  eliminated.  After  the  proper  conditions  were  secured, 
however,  no  further  difficulties  were  experienced,  and  the 
eighteen  tables  contain  all  of  the  measurements  that  were  taken. 

1  Since  there  is  considerable  lag  in  the  effect  produced  by  pressure  changes, 
the  changes  in  electromotive  force  measurements  do  not  correspond  to  these 
unless  the  changes  in  pressure  were  gradual. 


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21 


Discussion  of  Results 

To  facilitate  the  study  of  the  results  a  summary  of  the 
values  for  E,  Hi  and  E2  is  presented  in  Tables  II,  III  and  IV. 

TABLE  II — SUMMARY  OF  VALUES  FOR  E 


Calculated 

Measured 

Cone. 

Diff. 

New 
formula 

Old 
formula 

By  diff. 

By  sum. 

Average 

n/30 

n/5 

0.08530       0.08702 

0.08567 

0.08558 

0.08562 

n/6o    — 

n/3o 

0.03361 

0.03414 

0.03350 

0.03347 

0.03349 

n/go 

n/30 

0.05365 

0.05423    !    0.05334 

0-05333 

0-05334 

n/I2O  ~ 

n/30 

0.06744 

0.06828       0.06711 

0.06705 

0.06708 

w/150  - 

n/so 

0.07879 

0-07955    |    0.07847 

0.07847 

0.07847 

n/iSo  - 

n/30 

0.08778 

0.08858    \    0.08784 

0.08772 

0.08778 

n/3oo  - 

n/30      0.11305 

0.11400       0.11361 

0.11361 

0.11361 

In  Table  II  under  the  head  "Old  formula"  are  given 
the  values  calculated  from  the  regular  formula  for  concen- 
tration cells  without  boundary  potential,  namely  E  = 

2R.T       c 
P     In    \     In  Tables  III    and    IV   in    the   columns   headed 

"Old  formula,    Nc   =    826,"  are  given  the  values  calculated 
from  the  formulas  for  concentration  cells  with  boundary  po- 

1      T?         2RTNa  ,    d       A  ^         2RTNC  ,    d      _ 
tential,  namely,  £2  =       T?       In      and  Ei  =       ^       In  —  .     In 


these  calculations  R  was  taken  equal  to  8.3160  joules,  T 
equal  to  291.09°,  and  F  equal  to  96490.  The  conduc- 
tivity data  given  by  Goodwin  and  Haskell1  for  HC1  was 
plotted  and,  in  the  calculation  of  the  degree  of  dissociation 
at  the  various  concentrations  used  by  the  writer,  the  conduc- 
tivity values  were  taken  from  this  curve.  The  value  for 
infinite  dilution  was  taken  equal  to  380. 

A  comparison  of  the  measured  values  with  the  calculated 
in  these  three  tables  shows  fairly  close  agreement;  in  fact 
agreements  much  less  close  than  these  have  been  considered 
very  satisfactory  and  used  as  a  confirmation  of  the  accuracy 
of  the  formulas.  A  careful  inspection  of  these  tables,  however, 

1  Phys.  Rev.,  19,  386  (1904). 


22 


shows  that  the  calculated  values  are  in  every  instance  greater 
than  the  measured  and  this  was  found  to  be  the  case,  not  only 
for  these  results,  but  also  for  similar  measurements  made  by 
others. 

TABLE  III — SUMMARY  OF  VALUES  FOR  Ei 


Calculated 

Cone.  Diff. 

Measured 

Old  formula 

New  formula 

Nc  =  832 

Nc   =  826 

Nc  =  832 

Nc   =  826 

n/3o      -  n/5 

0.01397 

0.01463 

0.01512 

0.01433 

0.01484 

n/6o      -  n/30 

0.00571 

0.00580 

0.00594 

0.00571 

0.00585 

w/90      -  n/30 

0.00888 

0.00910 

0.00943 

0.00901 

0.00933 

n/I2O  n/3O 

0.01130 

0.01147 

0.01187 

0.01133 

0.01173 

n/i$o  —  n/30 

0.01315 

0.01337 

0.01384 

0.01324 

0.01371 

n/iSo  —  w/3o 

0.01475 

0.01488 

0.01541 

0.01474 

0.01527 

n/300  —  n/30 

0.01898 

0.01916 

0.01984 

0.01899 

0.01967 

TABLE  IV — SUMMARY  OF  VALUES  FOR  E2 


Calculated 

Cone.  Diff. 

Measured 

New  formula 

Old  formula 

Nc  =  832 

Nc   =  826 

Nc  =  832 

Nc   =   826 

n/30      -  n/s 

0.07163 

0.07140 

0.07046 

0.07282 

0.07188 

n/6o      -  n/so 

0.02773 

0.02789 

0.02776 

0.02833 

0.02820 

n/go      -  n/^o 

0.04446 

0.04463 

0.04431 

0.045  1  1 

0.04479 

n/i2o  —  n/3o 

0.05573 

0.05611 

0.05570 

0.05681 

0.05640 

w/i5o  —  n/3o 

0.06535 

0.06555 

0.06508 

0.06618 

0.06571 

n/iSo  —  w/3o 

0.07299 

0.07304 

0.07251 

0.07370 

0.07317 

w/300  —  w/30 

0.09488 

0.09406 

0.09338 

0.09484 

0.09416 

By  substituting  the  measured  values  in  the  formula  for 
the  electromotive  force  of  concentration  cells  without  boundary 
potential,  the  activity  ratios  for  the  various  concentration 
differences  were  calculated.  These  are  presented  in  Table 
V,  together  with  the  concentration  ratios  for  the  same  concen- 
tration differences  calculated  from  conductivity  measurements. 


TABLE  V — ACTIVITY  AND  CONCENTRATION  RATIOS 


Cone 

.  Diff. 

aiH 

CIH 

aiRCl 

CIHCI 

a2H 

<*a 

a2HC1 

^HCl 

n/5 
n/6o 
w/90 

W/I20 
H/I50 

H/lSo 

w/300 

-  w/30 
-  w/30 
—  w/30 
-  w/30 
-  w/30 
—  w/30 
-  n/30 

5-507 
1.949 

2-895 
3.806 

4-776 

5-750 
9.621 

5.664 

1-974 
2.946 

3.893 
4.880 

5-842 
9.687 

30-311 
3-799 
8-375 
14-455 
22.793 

33-037 
92.449 

12.881 
2.723 
4.784 
7  .080 
10.570 

13-443 
29.500 

This  table  again  brings  out  the  fact  that  the  activity  ratios 
for  the  ions  are  in  every  case  less  than  the  corresponding  con- 
centration ratios;  but  for  the  undissociated  part  of  the  acid 
the  activity  ratios  are  much  greater  than  the  concentration 
ratios,  and  as  the  dilution  becomes  greater  the  activity  in- 
creases more  rapidly  than  the  concentration. 

In  view  of  the  fact  that  for  all  non-electrolytes  and  even 
for  weak  electrolytes,  the  activity  has  been  shown  by  others 
to  be  proportional  to  the  concentration,  the  nature  of  these 
results  is  quite  remarkable.  They  confirm  the  statement  made 
in  the  consideration  of  the  preliminary  experiments,  that  either 
conductivity  measurements  do  not  give  true  values  for  the 
degree  of  dissociation  or,  else  the  concentration  of  the  ions  as 
well  as  the  undissociated  part  of  the  acid  is  not  proportional 
to  the  activity  in  the  case  of  hydrochloric  acid. 

It  will  be  remembered  that  Jahn,  in  his  measurements 
with  concentration  cells  having  boundary  potential,  found 
his  observed  values  to  be  less  than  those  calculated  from  the 
Nernst  equation.  Table  II  shows  the  same  relation  for  mea- 
surements with  concentration  cells  without  boundary  potential. 
To  explain  these  discrepancies  between  his  observed  and  cal- 
culated values,  Jahn  suggested  that  conductivity  is  not  an 
accurate  method  for  determining  the  degree  of  dissociation. 
He  doubted  the  validity  of  the  assumption  that  the  transport 
numbers  do  not  change  with  concentration,  and  this  assump- 
tion must  be  made  when  the  degree  of  dissociation  is  cal- 


24 


culated  from  conductivity  data.  It  will  be  remembered  that 
the  investigation  of  the  relation  between  transport  numbers 
and  concentration  is  one  of  the  primary  objects  of  this  paper. 
As  was  pointed  out  earlier  in  the  discussion  the  measure- 
ments in  this  work  give  two  new  and  distinct  methods  for  cal- 
culating transport  numbers.  One  is  by  the  formula  Nc  = 
E 


^  or  Nfl   = 


Tj* 

g2,    (one  serves  as  a  check  on  the  other)   and 
the    other    by    the    formula    for    boundary    potential    E  = 

(i  —  2NC)  -TJT"  'In  ~-     In  the  first  method  nothing  is  involved 
j?          c^ 

but  the  measurements  of  £2,  Ei  and  E  which  can  be  made 
with  a  very  high  degree  of  accuracy.  The  second  method, 
however,  involves  a  formula  which  is  possibly  questionable, 

particularly  the  ratio  ~ 

The  results  obtained  by  the  application  of  these  two  for- 
mulas are  presented  in  Table  VI. 

TABLE  VI — SUMMARY  OF  TRANSPORT  NUMBER  Nc 


Cone.  Diff. 

E! 

E 

<-t 

Average 

From  boundary 
E.  M.  F. 

Using  ^ 

C2 

Using01 

a-i 

w/30    —  n/s 
n/6o    —  w/30 
n/go    —  w/3o 

H/I2O  W/3O 

w/150  —  n/30 
n/iSo  —  w/3O 
w/300  —  w/30 

0.836 
0.828 

0.833 
0.830 

0-833 
0.831 

0.835 

0.8377 
0.830 

0.833 
0.831 
0.832 
0.8315 
0-833 

0.837 
0.829 

0.833 
0.830 
0.832 
0.831 
0.834 

0.831 
0.822 
0.827 
0.825 
0.830 
0.829 

0.832 

0.837 
0.829 

0-833 
0.831 

0-835 
0.831 

0-833 

Final  av.,  0.832 

It  should  be  noted  that  £2  and  Ei  give  practically  identical 
values  for  Nc  but  these  do  not  agree  with  the  values  calculated 
from  the  formula  for  boundary  potential.  According  to  the 
first  method  Nc  remains  constant  at  dilutions  greater  than 
n/9'o  and  increases  slightly  between  7^/30  and  n/$. 

In   Table  VI,   under  the  head  — ,  are  given   the   values 


25 

for  Nc  calculated  from  boundary  potentials  by  substituting 
the  activity  ratios  from  Table  V  in  place  of  concentration  ratios. 
This  gives  results  nearly  identical  with  those  given  by  the 
other  method.  One  of  the  most  important  features  of  these 
results  is  the  argument  they  add  to  the  view  that  the  formula 
for  calculating  boundary  potential  is  reliable  and  that  the 
formula  for  calculating  ionic  concentration  is  unreliable. 

To  facilitate  a  comparison  of  the  results  of  the  various 
investigators  who  have  measured  the  transport  numbers  of 
HC1  the  data  obtained  by  them  is  presented  in  Table  VII. 

All  of  the  recent  investigators  except  one  have  used  the 
Hittorf  method,  with  variations  to  eliminate  as  far  as  possible 
the  various  sources  of  error.  The  wide  variation  between  the 
different  results  of  each  investigator  as  well  as  the  lack  of 
agreement  between  the  values  of  different  investigators, 
shows  that  the  method  is  not  very  reliable.  Washburn1 
has  expressed  the  belief  that  during  such  experiments  water 
is  transported,  due  to  the  hydra tion  of  the  ions,  from  one  elec- 
trode chamber  to  the  other  when  the  solutions  are  fairly  con- 
centrated. Buchbock  also  came  to  the  same  conclusion. 
Their  evidence  appears  to  be  strong.  If  water  is  so  trans- 
ported then  it  is  evident  that  transference  numbers  cannot 
be  accurately  determined  by  this  method. 

The  moving  boundary  method  for  measuring  transfer- 
ence numbers  is  not  open  to  the  same  criticism,  but  so  few 
measurements  have  been  made  on  hydrochloric  acid  by  it 
that  scarcely  anything  can  be  said. 

The  two  methods  used  in  this  present  investigation  appear 
to  be  entirely  free  from  all  the  sources  of  error  attending 
the  Hittorf  method.  Their  limitations  have  been  discussed 
but  they  appear  to  give  more  consistent  and  reproducible  values 
than  haye  been  obtained  before. 

In  Table  VIII  is  presented  a  summary  of  the  measured 
values  for  boundary  potential  together  with  the  calculated 
values  when  Nc  is  given  the  commonly  accepted  value  0.826 
and  also  when  Nc  is  given  the  value  0.832  found  in  this  investi- 

1  Jour.  Am.  Chem.  Soc.,  31,  322  (1909). 


26 


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gation.  The  table  shows  much  better  agreement  between 
the  measured  values  and  those  calculated  with  Nc  =  0.826 
than  with  Nc  =  0.832.  This  is  really  a  false  agreement  as 
can  easily  be  shown.  In  the  discussion  of  transference  num- 
bers it  was  pointed  out  that  the  values  in  Table  VI,  in  the 

column    headed    -*    were    obtained    by    substituting    in    the 

formula  activity  ratios  in  the  place  of  concentration  ratios. 
This  gave  values  which  are  in  very  close  agreement  with  those 

determined    from    the   ratio   ^-      It    is    evident    then    that 

the  error  introduced  by  using  Nc  ••  •-  0.826  is  approximately 
counterbalanced  by  the  error  introduced  by  using  concentra- 
tion ratios  in  the  place  of  activity  ratios,  and  thus  causes  a 
false  agreement  between  measured  and  calculated  values. 

In  Table  VIII,  in  the  column  headed,  Nc  =  0.832  and 
—.  are  given  the  calculated  values  when,  for  N0  the  values 
measured  from  the  ratio  g1  are  used  and  for  --  are  used  the 
measured  activity  ratios.  The  remarkable  agreement  be- 
tween these  calculated  and  measured  values  speaks  for 
itself.  There  can  remain  little  doubt  concerning  the  exact- 
ness of  the  formula;  but  the  grave  necessity  for  accurate 
transference  values  and  activity  ratios  is  also  made  evident. 

TABLE  VIII — SUMMARY  OF  BOUNDARY  K.  M.  F. 


Cone.  Diff. 

Calculated 

Nc  =  0.826 

Nc  =  0.832 

Nc  =  0.832 

Measured 

and'-1 

and* 

and  -- 

C2 

C-i 

W/30 
w/6o 

-n/5 
—  w/30 

0.02837 
O.OIII3 

0.029301 
0.01126 

0.028931 

O.OIII2 

0.02883 
O.OIIOO 

n/go 

W/I20 

—  w/30 

0.01767                  O.OlSOO 
O.O2226                 O.O2267 

0.01771            0.01771 

0.02226                 0.02221 

w/i5o 

-w/30 

0.02593                  0.02641 

0.02605                 O.O26IO 

w/i8o 

—  w/30 

0.02887 

0.02941 

O.O29I4                 0.02912 

w/3oo 

-w/3o 

0.03715 

0.03783 

0.03777                 0.03785 

1  In  calculating  this  value  Nc  was  taken  equal  to  0.837. 


28 

In  the  discussion  of  Tables  II,  III  and  IV  it  was  pointed 
out  that  the  calculated  values  for  electromotive  forces  were 
invariably  greater  than  the  measured;  and  that  the  same  is 
true  of  all  similar  measurements  made  by  others.  The  most 
probable  explanation  for  this  is  some  constant  source  of  error 
in  the  formula.  One  such  source  has  been  pointed  out,  namely, 
the  formula  calls  for  an  activity  ratio,  but  in  place  of  this  a 
concentration  ratio,  calculated  from  conductivity  measure- 
ments, had  to  be  substituted.  After  a  careful  study  of  the 
most  probable  changes  within  the  cell,  it  appears  to  the  writer 
that  the  formula  does  not  exactly  fit  the  case.  The  following 
discussion  is  an  effort  to  make  this  point  evident  and  to  sug- 
gest an  improvement. 

The  formula  as  it  is  commonly  used  has  the  form 

W  =  EF  =  2RT  In  -. 
€2 

This  equation  is  supposed  to  represent  the  work  required  to 
transfer  one  electrochemical  equivalent  of  electricity  from  one 
electrode  to  the  other.  If  this  is  carried  out  in  a  double  cell 
so  that  there  can  be  no  change  at  the  boundary,  it  would  re- 
sult in  the  disappearance  of  one  mol  of  HC1  from  the  more 
concentrated  side  and  the  appearance  of  one  mol  in  the  dilute. 
The  work  involved  then  is  that  required  to  transfer  a  mol  of 
HC1  from  one  side  to  the  other.  In  case  of  complete  ioniza- 
tion  this  would  mean  the  transfer  of  one  equivalent  of  hydrogen 
ion  and  one  equivalent  of  chloride  ion,  and  would  be  represented 

by  the  expression  2RT  In  CJ.     This  is  the  same  as  the  third 

member  in  the  above  equation.  Under  the  conditions  of  com- 
plete dissociation  then  this  equation  is  entirely  applicable, 
but  in  no  actual  experiment  is  this  the  case.  What  would 
really  take  place  if  a  faraday  of  electricity  were  passed  through 
such  a  cell  would  be,  (i)  the  transfer  of  an  amount  of  chloride 
ion  equal  to  the  dissociation  of  the  acid,  (2)  the  transfer  of 
an  amount  of  hydrogen  ion  equal  to  the  dissociation  of  the  acid, 
(3)  the  transfer  of  an  amount  of  undissociated  acid  equal  to 
the  undissociated  part. 


29 

The  work  in  (i)  equals  the  work  in  (2)  and  may  be  repre- 
sented by 

Wi  =  «RT  In  C\ 

Cz 

The  work  in  (3)  may  He  represented  by 
W2  =  (i  —  a)  RT  In  Cl. 

C2 

Now 

CZH  =  c2a"  =  c2~ 
and 

c\H  =  c}ar  =  c\—^ 
and 

C2HC1  =  C2(i  —  a") 
and 

CIHCI  =  Ci(i  —  <*')• 

Combining  and  substituting, 

W  =  2«RT  /*  ^V  +  (i  -  a)  RT  In  ^  --  a£ 
Ciaf  ci(i  —  a') 

This  formula  is  not  absolutely  exact  because  of  the  un- 
certainty in  a.  The  assumption  is  made  that  the  dissociation 
is  the  same  in  both  sides,  but  this  evidently  is  not  true ;  so  the 

most  likely  value  to  use  for  a  is  - 

The  values  for  E,  Ei  and  Es  have  been  recalculated  ac- 
cording to  this  new  formula  and  are  recorded  in  Tables  II, 
III  and  IV  under  the  heading  "New  formula." 

In  Table  IX  are  presented  the  differences  (X  io5)  between 
the  measured  and  the  calculated  values  by  use  of  the  old  for- 
mula in  one  case  and  the  new  formula  in  the  other. 

It  is  evident  from  these  differences  that  the  new  formula 
is  an  improvement. 


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Conclusion 

From  all  this  it  is  concluded:  (i)  that  the  transport 
numbers  for  hydrochloric  acid  remain  constant  for  dilutions 
greater  than  n/^o,  and  thus  conductivity  measurements  give 
an  accurate  method  for  determining  ion  concentrations  in 
dilutions  greater  than  n/$o,  so  far  as  any  change  in  the  ve- 
locity of  the  ions  is  concerned;  (2)  that  the  concentration  is 
not  proportional  to  activity;  (3)  that  the  Nernst  equation  can- 
not be  applied  to  an  actual  concentration  cell  without  some 
changes;  (4)  that  the  formula  for  calculating  boundary  elec- 
tromotive force  gives  accurate  results  if  the  true  transport 
numbers  are  used  and  if  activity  is  substituted  in  the  place 
of  concentration.1 


1  Since  the  completion  of  this  investigation,  an  article  by  Duncan  A. 
Maclnnes  and  Karr  Parker  (Jour.  Am.  Chem.  Soc.,  37,  1445  (1915))  on  the 
KC1  concentration  cell  has  appeared  in  the  literature.  The  KC1  concentration 
cell  is  dealt  with  in  much  the  same  way  as  the  present  work  deals  with  the  HC1 
concentration  cell.  The  values  obtained  for  the  transport  numbers  show  prac- 
tically no  variation  with  concentrations  between  0.5  and  0.005  n.  It 
may  be  said,  in  general,  that  the  conclusions  arrived  at  in  this  investigation  of 
HC1  solutions  agree  remarkably  well  with  those  stated  by  Maclnnes  and  Parker 
for  KC1  solutions. 


THK  o_ 

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OVERDUE. 


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